Abstract

A ray-tracing approach is used to demonstrate efficient application of the
vectorial laws of reflection and
refraction to computational optics
problems. Both the full width at half-maximum (fwhm) and offset of Gaussian
beams resulting from off-center reflection and refraction are calculated for
spherical and paraboloidal surfaces of revolution. It is found that the
magnification and displacement depend nonlinearly on the miscentering. For these
geometries, the limits of accuracy of the lens approximation are examined
quantitatively. In contrast to the ray-tracing solution, this paraxial
approximation would predict a magnification of a beam’s fwhm that is
independent of miscentering, and an offset linearly proportional to the
miscentering. The focusing property of paraboloidal surfaces of revolution is
also derived in setting up the calculation.

References

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Table 1.

Absolute Magnification of fwhm of Beam by Reflection (Refraction in Final
Row) to Photodetector from Various Surfaces, in the Direction of
Displacement (y Magnification) (italics) and the
Perpendicular Direction
(x Magnification), by Exact Vectorial
Calculation (exact) or Approximation with the Lens Law or
Cross-Sectional Vectorial
Calculation (cross)a

Beam and Surface

Lens
Law

Vector
Calculation

No Displacement Magnification

1.5 mm Displacement

x Magnification

yMagnification

exact

cross

exact

cross

exact

cross

2 mm

Convex sphere

11.000

11.258

11.258

11.900

11.900

13.388

13.388

Concave sphere

−9.000

−9.226

−9.226

−9.788

−9.788

−11.094

−11.094

Concave paraboloid

−9.000

−9.143

−9.143

−9.482

−9.481

−10.209

−10.209

Convex sphere (refraction)

0.875

0.876

0.876

0.879

0.879

0.885

0.885

0.5 mm

Convex sphere

11.000

11.029

11.016

11.636

11.621

12.942

12.925

Concave sphere

−9.000

−9.021

−9.014

−9.552

−9.544

−10.696

−10.687

Concave paraboloid

−9.000

−9.016

−9.009

−9.345

−9.337

−10.028

−10.020

Convex sphere (refraction)

0.875

0.876

0.875

0.879

0.878

0.884

0.883

a The fwhm of the beam at the waist is 0.5 mm for the top half or
2.0 mm for the bottom half of the table. The remaining calculation
parameters are the same as in Fig. 4.

Table 2.

For Each Geometry, the Value of yC (the Displacement of the Surface Center
from the Laser Axis, in Millimeters) Is Shown, Where the fwhm Magnification
Calculation Gives the Corresponding Deviation
(5%, 10%, or 25%) in the
y Direction (top) or
x Direction (bottom) from the Lens-Law
Approximationa

Beam and Surface

yforΔy=5%

y for Δy=10%

yforΔy=25%

exact

cross

exact

cross

exact

cross

2 mm

Convex sphere

0.589

0.589

0.980

0.980

1.605

1.605

Concave sphere

0.554

0.554

0.936

0.936

1.553

1.553

Concave paraboloid

0.827

0.827

1.272

1.272

2.053

2.053

Convex sphere (refraction)

3.116

3.116

4.088

4.088

5.387

5.387

0.5 mm

Convex sphere

0.818

0.830

1.148

1.160

1.749

1.756

Concave sphere

0.790

0.799

1.117

1.117

1.699

1.699

Concave paraboloid

0.999

1.008

1.408

1.417

2.153

2.162

Convex
sphere (refraction)

3.170

3.207

4.161

4.179

5.514

5.523

Beam and Surface

yforΔx=5%

y for Δx=10%

yforΔx=25%

exact

cross

exact

cross

exact

cross

2 mm

Convex sphere

1.034

1.034

1.701

1.701

2.693

2.693

Concave sphere

0.972

0.972

1.635

1.635

2.616

2.616

Concave parabola

1.435

1.435

2.198

2.198

3.434

3.434

Convex sphere (refraction)

5.069

5.069

6.305

6.305

—

—

0.5 mm

Convex sphere

1.395

1.413

1.942

1.954

2.856

2.862

Concave sphere

1.354

1.372

1.890

1.899

2.789

2.789

Concave parabola

1.717

1.726

2.389

2.398

3.561

3.570

Convex sphere (refraction)

5.115

5.160

6.350

6.350

—

—

a Either the exact vector or cross-sectional approach is used. For example,
5% relative error means Mvector/Mlens=1.05. The remaining calculation parameters
are the same as in Table 1
and Fig. 4.

Tables (2)

Table 1.

Absolute Magnification of fwhm of Beam by Reflection (Refraction in Final
Row) to Photodetector from Various Surfaces, in the Direction of
Displacement (y Magnification) (italics) and the
Perpendicular Direction
(x Magnification), by Exact Vectorial
Calculation (exact) or Approximation with the Lens Law or
Cross-Sectional Vectorial
Calculation (cross)a

Beam and Surface

Lens
Law

Vector
Calculation

No Displacement Magnification

1.5 mm Displacement

x Magnification

yMagnification

exact

cross

exact

cross

exact

cross

2 mm

Convex sphere

11.000

11.258

11.258

11.900

11.900

13.388

13.388

Concave sphere

−9.000

−9.226

−9.226

−9.788

−9.788

−11.094

−11.094

Concave paraboloid

−9.000

−9.143

−9.143

−9.482

−9.481

−10.209

−10.209

Convex sphere (refraction)

0.875

0.876

0.876

0.879

0.879

0.885

0.885

0.5 mm

Convex sphere

11.000

11.029

11.016

11.636

11.621

12.942

12.925

Concave sphere

−9.000

−9.021

−9.014

−9.552

−9.544

−10.696

−10.687

Concave paraboloid

−9.000

−9.016

−9.009

−9.345

−9.337

−10.028

−10.020

Convex sphere (refraction)

0.875

0.876

0.875

0.879

0.878

0.884

0.883

a The fwhm of the beam at the waist is 0.5 mm for the top half or
2.0 mm for the bottom half of the table. The remaining calculation
parameters are the same as in Fig. 4.

Table 2.

For Each Geometry, the Value of yC (the Displacement of the Surface Center
from the Laser Axis, in Millimeters) Is Shown, Where the fwhm Magnification
Calculation Gives the Corresponding Deviation
(5%, 10%, or 25%) in the
y Direction (top) or
x Direction (bottom) from the Lens-Law
Approximationa

Beam and Surface

yforΔy=5%

y for Δy=10%

yforΔy=25%

exact

cross

exact

cross

exact

cross

2 mm

Convex sphere

0.589

0.589

0.980

0.980

1.605

1.605

Concave sphere

0.554

0.554

0.936

0.936

1.553

1.553

Concave paraboloid

0.827

0.827

1.272

1.272

2.053

2.053

Convex sphere (refraction)

3.116

3.116

4.088

4.088

5.387

5.387

0.5 mm

Convex sphere

0.818

0.830

1.148

1.160

1.749

1.756

Concave sphere

0.790

0.799

1.117

1.117

1.699

1.699

Concave paraboloid

0.999

1.008

1.408

1.417

2.153

2.162

Convex
sphere (refraction)

3.170

3.207

4.161

4.179

5.514

5.523

Beam and Surface

yforΔx=5%

y for Δx=10%

yforΔx=25%

exact

cross

exact

cross

exact

cross

2 mm

Convex sphere

1.034

1.034

1.701

1.701

2.693

2.693

Concave sphere

0.972

0.972

1.635

1.635

2.616

2.616

Concave parabola

1.435

1.435

2.198

2.198

3.434

3.434

Convex sphere (refraction)

5.069

5.069

6.305

6.305

—

—

0.5 mm

Convex sphere

1.395

1.413

1.942

1.954

2.856

2.862

Concave sphere

1.354

1.372

1.890

1.899

2.789

2.789

Concave parabola

1.717

1.726

2.389

2.398

3.561

3.570

Convex sphere (refraction)

5.115

5.160

6.350

6.350

—

—

a Either the exact vector or cross-sectional approach is used. For example,
5% relative error means Mvector/Mlens=1.05. The remaining calculation parameters
are the same as in Table 1
and Fig. 4.